A Double-Track Auction for Substitutes and Complements

We propose a new t^atonnement process called a double-track auction for efficiently allocating multiple heterogeneous indivisible items in two distinct sets S1 and S2 to many buyers who view items in the same set as substitutes but items across the two sets as complements. The auctioneer initially announces sufficiently low prices for items in one set, say S1, but sufficiently high prices for items in the other set S2. In each round, the buyers respond by reporting their demands at the current prices and the auctioneer adjusts prices upwards for items in S1 but downwards for items in S2 based on buyers' reported demands until the market is clear. Unlike any existing auction, this auction is a blend of a multi-item ascending auction and a multi-item descending auction. We prove that the auction finds an efficient allocation and its market-clearing prices in finitely many rounds. Based on the auction we also establish a dynamic, efficient and strategy-proof mechanism.